Abstract
In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish various changes of scale formulas for the conditional transforms and the conditional convolution products.
1. Introduction
Let denote the Wiener space, the space of real-valued continuous functions on the interval with . It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation and under translations on [1, 2]. Yoo and his coauthors [3] presented a change of scale formula for Wiener integrals of functions on the abstract Wiener space [4] which generalizes . We note that the functions used in [3] are the products of generalized cylinder functions on and the functions on the Fresnel class [5] which is the space of Fourier-Stieltjes transforms of measures on a separable real Hilbert space densely embedded in , and note that they need not be bounded or continuous.
Let denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [6]. Two kinds of integral transforms, which are known as an analytic conditional Fourier-Feynman transform and a conditional convolution product on , were introduced by the author and his coauthors [7–9] using some conditioning functions. In fact, the authors [9] investigated the conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions on and established various relationships that occur among them using the conditioning function given by , where . Moreover they derived several changes of scale formulas for the conditional transforms and the conditional convolution products, which simplify the evaluations of the conditional expectations, because the probability measure used on may not be scale-invariant. We note that has no drift and contains the present position of the path . In [7], Cho extended the results in [9] using a more generalized stochastic process given by for and , where denotes the Paley-Wiener-Zygmund stochastic integral [6], and a.e.) and are of bounded variation and absolutely continuous, respectively, on . Here he used the conditioning function for which generalizes , and the functions and have the effect that the normal density defining the premeasure of cylinder sets on contains the mean function and the variance function , respectively. Using the conditioning function for , Cho [8] also derived similar results in [7]. We note that contains the present positions of generalized Wiener paths and does not; that is, the conditional expectation at generalizes the probability of the Wiener paths which pass through at time for while describes the probability of the Wiener paths which pass through at time for . In other words, describes the conditional expectation of Wiener paths which pass through at each past time and at the present time , while only describes the conditional expectation of Wiener paths which pass through at each past time .
In this paper, using two simple formulas for conditional expectations over paths [10, 11], we evaluate conditional expectations of the products of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. Then we investigate their relationships. In particular, we establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and changes of scale formulas, we use multivariate normal distributions so that the evaluation formulas are unaffected by the present positions of paths, despite the existences of the present positions of the paths in the conditioning functions. We also note that the change scale formulas in [7–9] are expressed by finite sums, while the formulas in this paper are expressed by a limit with a complete orthonormal set of . Moreover, the conditional Fourier-Feynman transforms and convolution products of cylinder functions in [7–9] are still cylinder functions while the transforms and the convolutions of the functions in this paper are not cylinder functions. In fact, with the conditioning functions and , we evaluate conditional expectations, namely, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by where is a complex Borel measure of bounded variation on , with , and is an orthonormal subset of . Then we show that the conditional Fourier-Feynman transform of the conditional convolution product of the functions defined by where is a complex Borel measure of bounded variation on , , can be expressed by the formula for a nonzero real , almost surely , and almost surely , where and is the probability distribution of on the Borel class of . Moreover, replacing by , we can recover the same formula with and the effects of drift will be investigated on the polygonal function of so that the results do not depend on a particular choice of the initial distribution of the paths. We also note that the functions in (1) extend the initial state of the Schrödinger equation [12].
2. A Function Space and Preliminary Results
For a positive real , let denote the space of real-valued continuous functions on the time interval with the supremum norm and let be the analogue of Wiener measure according to the probability measure on the Borel class of [6].
Let and denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let be integrable and let be a random vector on assuming that the value space of is a normed space with the Borel -algebra. We will adopt the conditional expectation of given , which is described in [8, 13, 14].
For an extended real number with , suppose that and are related by (possibly if ). Let . For let be a measurable function on such that . Then we write .
Define a stochastic process by for and , where denotes an indicator function. For , let and, for any function on , define the polygonal function of by for . For , define the polygonal function of by the right-hand side of (4), where is replaced by . If , is interpreted as with . For and let For , , and any nonsingular positive matrix on , let where denotes the dot product on .
For a function , let , let , and let for and . Suppose that exists, where the expectation is taken over the first variable. By the same method as in Theorem 2.5 of [10] we have for a.e. , and for a.e. where , the expectation is taken over the variable and , are the probability distributions of , on , , respectively. Let and be the right-hand sides of (7) and (8), respectively. If, for a.e. and a.e. , has an analytic extension on , then it is called a generalized analytic conditional Fourier-Wiener transform of given with the parameter and is denoted by Moreover if has a point-wise limit as approaches through , then it is called a generalized -analytic conditional Fourier-Feynman transform of given with the parameter and is denoted by For , we define a generalized -analytic conditional Fourier-Feynman transform of given by the formula and are similarly defined with .
Let be defined on . For and , suppose that exists. By the same method as in Theorem 2.5 of [10] we have Let and be the right-hand sides of (12) and (13), respectively. If has an analytic extension on , then it is called a generalized conditional convolution product of and given with the parameter and denoted by Moreover if has a point-wise limit as approaches through , then it is denoted by and are similarly understood with .
For , let , let be the subspace of generated by , and let be the orthogonal complement of . Let be the projection given by and let be the orthogonal projection.
The following lemmas are essential to prove the results in the next sections [8].
Lemma 1. Let . Then for a.e. we have where is the multiplication operator defined by for
Lemma 2. Let and . Then we have where and denotes the inner product on .
Lemma 3. Let be a subset of such that is an independent set. Then the random vector has the multivariate normal distribution with mean vector and covariance matrix . Moreover, for any Borel measurable function we have where means that if either side exists, then both sides exist and they are equal.
3. Generalized Conditional Fourier-Feynman Transforms
In this section we derive the conditional Fourier-Feynman transforms of functions on and investigate the effects of the drift .
Let , let be any fixed positive integer, and let be an orthonormal subset of such that both and , are independent sets. Let be the space of cylinder functions having the form for a.e. , where and , Without loss of generality we can take to be Borel measurable.
Let be the class of all -valued Borel measures of bounded variation over and let be the space of all functions which for have the form for a.e. . Note that is a Banach algebra [6].
Let be the orthonormal set obtained from , , by the Gram-Schmidt orthonormalization process. Keeping the orders of the elements in and , respectively, let be the matrix of change of coordinates from to . For , , and , let for , let , let , and let .
Theorem 4. Let for a.e. , where and are given by (19) and (20), respectively. Then we have for , a.e. and a.e. where is the identity matrix and, for , and for Furthermore for . If , then, for a nonzero real , is given by the right-hand side of (21) with replacing by . In this case, for .
Proof. For , a.e. and a.e. , we have by Lemma 1, where For , let if , where By Lemma 3 we have where and . By simple calculations we have Note that if , then we have by the change of variable theorem and for we have by the Bessel inequality. By the Morera theorem with aid of the Hölder inequality and the dominated convergence theorem, exists for . For we have by Lemma 3 and the Young inequality [15] so that because . If , then the final results follow from the dominated convergence theorem.
Remark 5. For let for . Let be the matrix of change of coordinates from to . Now for , so that for and we have and . Now the expression in and can be replaced by .
By Lemma 1.1 in [16] and (27) with minor modifications, we have the following theorem.
Theorem 6. Let be as given in Theorem 4 with , and let . Then for and a.e. , we have for .
Theorem 7. Let the assumptions be as given in Theorem 6. Suppose that is concentrated on the set . Then is given by the right-hand side of (21), where is replaced by and for a.e. and . In this case, for .
Proof. For a.e. , , and , we have so that it is not difficult to show Now is bounded by which is finite from Lemma 1.1 in [16]. By Lemma 3, Theorem 4, and the change of variable theorem, we have for which converges to as approaches through by Lemma 1.2 in [16].
For let , let , , and let . For and , let
Theorem 8. Suppose that the assumptions and notations are as given in Theorem 4. Let . Then we have for , a.e. and a.e. where and are given by (22) and (32), respectively. Furthermore , for . If , then, for a nonzero real , , is given by the right-hand side of (33) with replacing by . In this case, for .
Proof. For let with . For , a.e. and a.e. we have by Lemma 2For we have by the Bessel inequality and by the Schwarz inequality. By the Morera theorem we establish the first part of the theorem. If , then the final results follow from the dominated convergence theorem.
Theorem 9. Let be as given in Theorem 4 with , and let . Then for and a.e. , we have for .
Proof. For , is bounded by from (35), which is finite by Lemma 1.1 in [16].
Because if and only if , we can prove the following theorem using the same method as used in the proof of Theorem 7.
Theorem 10. Let the assumptions be as given in Theorem 7. Suppose that for . Then for a nonzero real and a.e. , is given by the right-hand side of (33), where is replaced by and , for a.e. and . In this case, for .
4. Generalized Conditional Convolution Products
In this section we derive the conditional convolution products of the functions in the previous sections and investigate the effects of the drift .
Theorem 11. Let , and , be related by (19), respectively, where and with . Let and be related by (20), respectively. Furthermore let , for a.e. and let , . Then we have for , a.e. and a.e. where is given by (22). Moreover , where if either or , if and if .
Proof. For , a.e. and a.e. , we have Using the same method as used in the proof of Theorem 4, we havewith replacing by in and . Now for we have formally By the same method as used in the proof of Theorem 3.2 in [9], we have with the existence of for if either or . Using the same methods as used in the proof of Theorem 3.2 in [9] again, we have the remainder part of the theorem.
Theorem 12. Let the assumptions be as given in Theorem 11. Then we have for , a.e. and a.e , where and are given by (22) and (32), respectively. Moreover , , where if either or , if and if .
Proof. Replacing by in the proof of Theorem 8 with Theorem 11, we have for , , and Now for , the formal expression is bounded by By the same process as used in the proof of Theorem 11, the proof is completed.
Using the fact and the same method as used in the proof of Theorem 3.3 in [9], we can prove the following two theorems.
Theorem 13. Let and be as given in Theorem 11. Let be a nonzero real number. Then for or , and , we have the following: (1)if , then for (2)if and , then for (3)if and , then for (4)if , then for
Theorem 14. If we replace by in Theorem 13, then the conclusions of the theorem are still true, where is replaced by .
Remark 15. The formal expressions in each theorem of this section and of the previous sections include the fact that there exists a cylinder function with for a.e. .
5. Relationships between Transforms and Convolution Products
In this section, we investigate the inverse transforms of the conditional Fourier-Feynman transforms of the functions as given in the previous sections. We also show that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed as the products of the analytic conditional Fourier-Feynman transforms of each function.
Theorem 16. Let be a nonzero real number. Then, under the assumptions as given in Theorem 7, we have for a.e. for , and for for a.e. , as approaches through .
Proof. For , a.e. and a.e. , we have by the change of variable theorem, Theorems 4 and 7where . Let . Then we have by Lemma 3 and the change of variable theoremLetting through which satisfies , we have the first part of the theorem by Theorem 1.18 in [17]. If , then the remainder part of the theorem follows from Theorem 1.25 in [17].
Theorem 17. Let be a nonzero real number. Then, under the assumptions as given in Theorem 10, we have for a.e. for , and for for a.e. , as approaches through .
Proof. For , a.e. and a.e. , we have by the change of variable theorem and Theorems 8 and 10where and , . Let . Then we have by Lemma 3, the Minkowski inequality, and the change of variable theorem Letting through which satisfies and , we have the first part of the theorem by Theorem 1.18 of [17]. If , then the remainder part of the theorem follows from Theorem 1.25 of [17].
Theorem 18. Under the assumptions as given in Theorem 11, we have for , a.e. and a.e.
Proof. For , the existence of follows from Theorems 4 and 11. We have for a.e. and a.e. Let and . Then we haveby the change of variable theorem. Now we have so that we have by Theorem 4which is the desired result.
Theorem 19. Under the assumptions as given in Theorem 12, we have for , a.e. and a.e.
Proof. For , the existence of follows from Theorems 8 and 12. We have for a.e. and a.e. Let and . By the change of variable theorem we have using the same process as used in the proof of Theorem 18. From a long calculation it is not difficult to show so that we have by Theorem 8which completes the proof.
By Theorems 4, 7, 8, 10, 13, 14, 18, and 19, we have the following two theorems.
Theorem 20. Let and be as given in Theorem 11, where . Let be a nonzero real number. Then we have for a.e. for a.e. , and for a.e.
Theorem 21. Let the assumptions be as given in Theorem 20 except for . Moreover suppose that is concentrated on . Then we have In addition, suppose that for . Then we have
6. Change of Scale Formulas for the Transforms and Convolutions
In this section we investigate change of scales for evaluating the conditional expectations of the functions as described in the previous sections.
Let be a complete orthonormal basis of containing . For , , and let . Let be a nonzero real number and let be a sequence in converging to as approaches .
Theorem 22. Let and let be as given in Theorem 4. Then we have for , a.e. and a.e. Moreover if , then is given by the right-hand side of (66) with replacing by .
Proof. For , and a.e. we have by the same process as used in the proof of Theorem 4where and is given by (22). By a simple calculation we have which holds for by an analytic continuation. Now, by the dominated convergence theorem and the Parseval identity, we have (66). The remainder part of the theorem follows from the dominated convergence theorem.
Using a similar process as used in the proofs of Theorems 8, 22, and 23, we can prove the following theorems by the dominated convergence theorem.
Theorem 23. Let and let be as given in Theorem 4. Then we have for , a.e. and a.e. where . Moreover if , then is given by the right-hand side of (69) with replacing by .
Theorem 24. Let the assumptions be as given in Theorem 11. Then we have for , a.e. and a.e. Moreover if , then is given by the right-hand side of (70) with replacing by .
Theorem 25. Let the assumptions be as given in Theorem 11. Then we have for , a.e. and a.e. where . Moreover if , then , is given by the right-hand side of (71) with replacing by .
Remark 26. (1)In the evaluation of for , the scale in is moved into on (66). This is the reason why (66) is called a change of scale formula for the transform.(2)All the results of this paper are independent for a particular choice of the initial distribution .(3)The results of this paper generalize most of theorems in [8, 9, 13, 18].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research work of Cho was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education (2017R1D1A1B03029876).